15 July 2002
Physics Letters A 299 (2002) 650–655
www.elsevier.com/locate/pla
High-temperature superconductivity and charge segregation in
a model with strong long-range electron–phonon and
Coulomb interactions
A.S. Alexandrov a , P.E. Kornilovitch b,∗
a Department of Physics, Loughborough University, Loughborough LE11 3TU, UK
b Hewlett Packard Labs, 1501 Page Mill Road, Palo Alto, CA 94304, USA
Received 13 March 2002; received in revised form 10 May 2002; accepted 22 May 2002
Communicated by A.R. Bishop
Abstract
An analytical method of studying strong long-range electron–phonon and Coulomb interactions in complex lattices is
presented. The method is applied to a perovskite layer with anisotropic coupling of holes to the vibrations of apical atoms.
Depending on the relative strength of the polaronic shift Ep and the inter-site Coulomb repulsion Vc , the system is either
a polaronic Fermi liquid, Vc > 1.23Ep , a bipolaronic superconductor, 1.16Ep < Vc < 1.23Ep , or a charge segregated insulator,
Vc < 1.16Ep . In the superconducting window, the carriers are mobile bipolarons with a remarkably low effective mass. The
model describes the key features of the underdoped superconducting cuprates.  2002 Elsevier Science B.V. All rights reserved.
PACS: 74.20.Mn; 71.38.-k; 71.38.Mx
There is clear experimental [1–5] and theoretical [6–15] evidence for strong electron–phonon (el–
ph) interaction in high-Tc superconducting cuprates
(HTSC). Electron correlations are also important in
shaping the Mott–Hubbard insulating state of parent undoped compounds [16]. The theory of high-Tc
cuprates must treat both interactions on equal footing as was suggested some time ago [6]. In recent
years many publications addressed the fundamental
problem of competing el–ph and Coulomb interactions in the framework of the Holstein–Hubbard model
* Corresponding author.
E-mail address: pavel_kornilovich@hp.com
(P.E. Kornilovitch).
[11–15] where both interactions are short-range (onsite). The mass of bipolaronic carriers in this model is
very large and the critical temperature is suppressed
down to a Kelvin scale. However, in the cuprates the
screening is poor so that the el–ph interaction necessarily has to be long-range. Motivated by this fact, we
have proposed that a long-range Fröhlich, rather than
short-range Holstein, interaction should be the adequate model for the cuprates [17,18]. Differently from
the usual continuum Fröhlich model (for review see
[6,7]), we introduced a multipolaron Fröhlich-like lattice model with electrostatic forces fully taking into
account the discreteness of the lattice, finite electron
bandwidth, and the quantum nature of phonons. A single small polaron with the Fröhlich interaction was
discussed long time ago [19]. Analytical [17] and ex-
0375-9601/02/$ – see front matter  2002 Elsevier Science B.V. All rights reserved.
PII: S 0 3 7 5 - 9 6 0 1 ( 0 2 ) 0 0 6 8 4 - 9
A.S. Alexandrov, P.E. Kornilovitch / Physics Letters A 299 (2002) 650–655
act Monte-Carlo [18] studies of the simple chain and
plane lattices with a long-range el–ph coupling revealed a several-order lower effective mass of this polaron than that of the small Holstein polaron. Later,
the polaron and bipolaron cases of the chain model
were analyzed in more detail in Refs. [20] and [21],
confirming low masses of both types of carriers. Qualitatively, a long-range el–ph interaction results in a
lighter mass because the extended lattice deformation
changes gradually as the carrier moves through the lattice.
In this Letter, we study a realistic multi-polaron
model of the copper–oxygen perovskite layer which is
the major structural unit of the HTSC compounds. The
model includes the infinite on-site repulsion (Hubbard
U term), long-range inter-hole Coulomb repulsion Vc ,
and long-range Fröhlich interaction between in-plane
holes and apical oxygens. We find that, within a certain window of Vc , the holes form inter-site bipolarons
with a remarkably low mass. The bipolarons repel and
the whole system is a superconductor with a high critical temperature. At large Vc , the system is a polaronic
Fermi liquid and at small Vc it is a charge segregated
insulator.
To deal with the model’s considerable complexity
we first describe a theoretical approach that makes
the analysis of complex lattices simple in the strong
coupling limit. The model Hamiltonian explicitly includes long-range electron–phonon and Coulomb interactions as well as kinetic and deformation energies.
An implicitly present infinite Hubbard term prohibits
double occupancy and removes the need to distinguish
fermionic spins. Introducing fermion operators cn and
phonon operators dmα , the Hamiltonian is written as
T (n − n )cn† cn − Vc (n − n )cn† cn cn† cn
H =−
n=n
−ω
gα (m − n)(emα · um−n )
†
× cn† cn dmα
+ dmα
1
†
+ω
dmα +
dmα
.
2
mα
nm
651
between m and n.) We assume that all the phonon
modes are dispersionless with frequency ω and that the
electrons do not interact with displacements of their
own atoms, gα (0) ≡ 0. We also use h̄ = 1 throughout
the Letter.
In the limit of strong el–ph interaction, it is convenient to perform the Lang–Firsov
canonical transformation [22]. Introducing S = mnα gα (m − n)(emα ·
†
− dmα ) one obtains a transformed
um−n )cn† cn (dmα
Hamiltonian without an explicit el–ph term:
H̃ = e−S H eS
1
†
dmα
=−
σ̂nn cn† cn + ω
dmα +
2
mα
n=n
+
v(n − n )cn† cn cn† cn − Ep
cn† cn .
n=n
(2)
n
The last term describes the energy which polarons gain
due to el–ph interaction. Ep is the familiar polaron
(Franck–Condon) shift,
Ep = ω
(3)
gα2 (m − n)(emα · um−n )2 ,
mα
which we assume to be independent of n. Ep is a natural measure of the strength of the el–ph interaction.
The third term in Eq. (2) is the polaron–polaron interaction:
v(n − n ) = Vc (n − n ) − Vpa (n − n ),
Vpa (n − n ) = 2ω
(4)
gα (m − n)gα (m − n )
mα
× (emα · um−n )(emα · um−n ),
(5)
where Vpa is the inter-polaron attraction due to joint
interaction with the same vibrating atoms. Finally, the
first term in Eq. (2) contains the transformed hopping
operator σ̂nn :
gα (m − n)(emα · um−n )
σ̂nn = T (n − n ) exp
mα
(1)
Here emα is the polarization vector of αth vibration
coordinate at site m, um−n ≡ (m − n)/|m − n| is the
unit vector in the direction from electron n to ion m,
and gα (m − n) is a dimensionless el–ph coupling
function. (gα (m − n) is proportional to a force acting
− gα (m − n )(emα · um−n )
†
(6)
− dmα .
× dmα
At large Ep /T (n − n ) this term is a perturbation.
In the first order of the strong coupling perturbation
theory [6], σ̂nn should be averaged over phonons
652
A.S. Alexandrov, P.E. Kornilovitch / Physics Letters A 299 (2002) 650–655
because there is no coupling between polarons and
phonons in the unperturbed Hamiltonian (the last three
terms in Eq. (2)). For temperatures lower than ω, the
result is
t (n − n ) ≡ σ̂nn ph
= T (n − n ) exp −G2 (n − n ) ,
G2 (n − n ) =
(7)
gα (m − n)(emα · um−n )
mα
× gα (m − n)(emα · um−n )
− gα (m − n )(emα · um−n ) .
(8)
By comparing Eqs. (3), (5), and (8), the mass renormalization exponents can be expressed via Ep and Vpa
as follows:
1
1
2
Ep − Vpa (n − n ) .
G (n − n ) =
(9)
ω
2
This is the simplest way to calculate G2 and (bi)polaron masses once the ‘static’ parameters Ep and Vpa are
known.
It is easy to see from the above equations that the
long-range el–ph interaction increases Ep and Vpa but
reduces G2 (when measured in natural units of Ep /ω).
Thus polarons get tighter and at the same time lighter.
Bipolarons form when Vpa exceeds Vc and they are
relatively light too. We note that the Holstein model
is the limiting case with the highest possible G2 =
Ep /ω. In this respect, the Holstein model is not a
typical el–ph model.
To obtain analytical description of the multi-polaron system we restrict our consideration to the strong
coupling case |v| t. In this regime the polaron
kinetic energy is the smallest energy and thus can be
treated as a perturbation. The system is adequately
described by a purely polaronic model:
Hp = H0 + Hpert ,
H0 = −Ep
n
Hpert = −
n=n
cn† cn +
(10)
n=n
v(n − n )cn† cn cn† cn , (11)
t (n − n )cn† cn .
(12)
Fig. 1. Four octahedra of the copper–oxygen perovskite layer. Holes
reside on the in-plane oxygens but interact with apical oxygens.
The many-particle ground state of H0 depends on
the sign of the polaron–polaron interaction, the carrier density, and the lattice geometry. Here we consider a two-dimensional lattice of ideal octahedra that
can be regarded as a simplified model of the copper–
oxygen perovskite layer, see Fig. 1. The lattice period is a = 1 and the distance between the apical sites
and the central plane is h = a/2 = 0.5. The hole degrees of freedom in the cuprates are the oxygen pstates. We assume that all in-plane atoms, both copper
and oxygen, are static but apical oxygens are independent three-dimensional isotropic harmonic oscillators.
Thus there are six lattice degrees of freedom per cell.
Because of poor screening the hole–apical interaction
is purely Coulombic,
gα (m − n) = κα /|m − n|2 ,
α = x, y, z.
To account for the experimental fact that the holes
couple stronger to z-polarized phonons
√ than to the
others [3], we choose κx = κy = κz / 2. The direct
hole–hole repulsion is
√
Vc / 2
Vc (n − n ) =
|n − n |
so that the repulsion between two holes in the NN
configuration is Vc . We also include the bare nearest
neighbor (NN) hopping TNN , the next nearest neighbor
(NNN) hopping across copper TNNN , and the NNN
hopping between octahedra TNNN
.
A.S. Alexandrov, P.E. Kornilovitch / Physics Letters A 299 (2002) 650–655
653
According to Eq. (3), the polaron shift is given by
the lattice sum (after summation over polarizations):
1
h2
2
+
Ep = 2κx ω
|m − n|4 |m − n|6
m
= 31.15κx2ω,
(13)
where the factor 2 accounts for the two layers of
apical sites. (For reference, Cartesian coordinates
are n = (nx + 1/2, ny + 1/2, 0), m = (mx , my , h),
nx , ny , mx , my being integers.) The polaron–polaron
attraction is
Vpa (n − n ) = 4ωκx2
h2 + (m − n ) · (m − n)
m
|m − n |3 |m − n|3
.
(14)
Performing lattice summations for the NN, NNN,
and NNN configurations one finds Vpa = 1.23Ep ,
0.80Ep , and 0.82Ep , respectively. Substituting these
results in Eqs. (4) and (9) we obtain the full interpolaron
√ interaction: vNN = Vc√− 1.23Ep , vNNN =
Vc / 2 − 0.80Ep , vNNN
= Vc / 2 − 0.82Ep , and the
mass renormalization exponents: G2NN = 0.38(Ep /ω),
G2NNN = 0.60(Ep /ω), G2
NNN = 0.59(Ep /ω).
Let us now discuss different regimes of the model.
At Vc > 1.23Ep , no bipolarons are formed and the
systems is a polaronic Fermi liquid. The polarons
tunnel in the square lattice with NN hopping t =
TNN exp(−0.38Ep /ω) and NNN hopping t = TNNN ×
exp(−0.60Ep /ω). (Since G2NNN ≈ G2
NNN one can neglect the difference between NNN hoppings within
and between the octahedra.) The single-polaron spectrum is therefore
E1 (k) = −Ep − 2t [cos kx + cos ky ]
± 4t cos(kx /2) cos(ky /2).
(15)
The polaron mass is m∗ = 1/(t + 2t ). Since in general
t > t , the mass is mostly determined by the NN
hopping amplitude t.
While the infinite Hubbard U prevents the simplest
on-site bipolaron the coupling to apical oxygens allows an inter-site NN bipolaron if Vc < 1.23Ep . The
inter-site bipolarons tunnel in the plane via four resonating (degenerate) configurations A, B, C, and D,
see Fig. 2. In the first order in Hpert , one should retain only these lowest energy configurations and discard all the processes that involve configurations with
Fig. 2. Top view on the perovskite layer. The apical sites are not
shown. The four bipolaron configurations A, B, C, and D all have
the same energy. Some possible single-polaron hoppings t are
indicated by arrows. Note that the bipolaron movement is first-order
in t .
higher energies. The result of such a projection is the
bipolaronic Hamiltonian
Hb = (Vc − 3.23Ep )
†
×
Al Al + Bl† Bl + Cl† Cl + Dl† Dl
l
−t
A†l Bl + Bl† Cl + Cl† Dl + Dl† Al + h.c.
l
−t
n
†
†
†
Cl + Cl+x
Dl + Dl−y
Al
A†l−x Bl + Bl+y
+ h.c. ,
(16)
where l numbers octahedra rather than individual sites,
x = (1, 0), and y = (0, 1). A Fourier transformation
and diagonalization of a 4 × 4 matrix yields the
bipolaron spectrum:
E2 (k) = Vc − 3.23Ep
± 2t cos(kx /2) ± cos(ky /2) .
(17)
There are four bipolaronic subbands combined in
a band of width 8t . The effective mass of the lowest
band is m∗∗ = 2/t . The bipolaron binding energy
is ∆ = 2E1 (0) − E2 (0) = 1.23Ep − Vc − 8t − 4t .
Because of an infinite Hubbard repulsion, the energy
splitting between the singlet and triplet inter-site bipolaron states is zero.
654
A.S. Alexandrov, P.E. Kornilovitch / Physics Letters A 299 (2002) 650–655
We would like to emphasize that the inter-site bipolaron moves already in the first order in polaron
hopping. This remarkable property is entirely due to
the strong on-site repulsion and long-range electron–
phonon interaction that leads to a nontrivial connectivity of the lattice. This situation is unlike all other models studied previously. (Usually the bipolaron moves
only in the second order in polaron hopping and therefore is very heavy.) In our model, this fact combines
with a weak renormalization of t yielding a superlight
bipolaron with mass m∗∗ ∝ exp(0.60Ep /ω). We recall
that in the Holstein model m∗∗ ∝ exp(2Ep /ω). Thus
the mass of the Fröhlich inter-site bipolaron scales approximately as cubic root of that of the Holstein onsite bipolaron.
At even stronger el–ph interaction, Vc < 1.16Ep ,
NNN bipolarons become stable. More importantly,
holes can now form 3- and 4-particle clusters. The
dominance of the potential energy over kinetic in
Hamiltonian (10) enables us to readily investigate
these many-polaron cases. Three holes placed within
one oxygen square have four degenerate
states with
√
energy 2(Vc − 1.23Ep ) + Vc / 2 − 0.80Ep . The
first-order polaron hopping processes mix the states
resulting in a ground state linear combination
with
√
energy E3 = 2.71Vc − 3.26Ep − 4t 2 + t 2 . It is
essential that between the squares such triads could
move only in higher orders in polaron hopping. In the
first order, they are immobile. A cluster of four holes
has only one state within a square of oxygen atoms. Its
energy is
Vc
E4 = 4(Vc − 1.23Ep ) + 2 √ − 0.80Ep
2
= 5.41Vc − 6.52Ep .
This cluster, as well as all the bigger ones, is also
immobile in the first order of polaron hopping. We
conclude that at Vc < 1.16Ep the system becomes
a charge segregated insulator within the first-order perturbation theory with respect to the kinetic energy.
A more accurate characterization of the charge segregated state and the phase boundary requires higherorder corrections.
The superconductivity window that we have found,
1.16Ep < Vc < 1.23Ep , is quite narrow. The fact,
that within this window there are no three or higher
polaron bound states, implies that bipolarons repel
each other. The system is effectively the charged
Bose gas, which is a well known superconductor (for
a review, see Ref. [6]). It follows from our model
that superconductivity in cuprates should be very
sensitive to any external factor that affects the balance
between Vc and Ep . For instance, pressure changes the
octahedra geometry and hence Ep and Vpa . Chemical
doping enhances internal screening and consequently
reduces Ep .
We now assume that the superconductivity condition is satisfied and show that our ‘Fröhlich–Coulomb’
model possesses many key properties of the underdoped cuprates. The bipolaron binding energy ∆
should manifest itself as a normal state pseudogap with size of approximately half of ∆ [6]. Such
a pseudogap has indeed been observed in many cuprates. There should be a strong isotope effect
√ on
the (bi)polaron mass because t, t ∝ exp(−const M).
Therefore the replacement of O16 by O18 increases
the carrier mass [24]. Such an effect has been observed in the London penetration depth of the isotopesubstituted samples [1]. The mass isotope exponent,
αm = d ln m∗∗ /d ln M, was found to be as large as
αm = 0.8 in La1.895 Sr0.105 CuO4 . Our theoretical exponent is αm = 0.3Ep /ω, so that the bipolaron mass
enhancement factor is exp(0.6Ep /ω) 5 in this material. With the bare hopping integral TNNN = 0.2 eV
we obtain the in-plane bipolaron mass m∗∗ 10me .
Calculated with this value the in-plane London penetration depth,
1/2
λab = m∗∗ /8πne2
316 nm
(n is the hole density), agrees well with the measured
one λab 320 nm. Taking into account the c-axis
tunneling of bipolarons, the critical temperature of
their Bose–Einstein condensation can be expressed in
terms of the experimentally measured in-plane and caxis penetration depths, and the in-plane Hall constant
RH as
1/3
.
Tc = 1.64f eRH /λ4ab λ2c
Here f ≈ 1, and Tc , eRH and λ are measured in K,
cm3 and cm, respectively [25]. This expression neglects both the hard-core and long-range electrostatic
repulsion of bipolarons. The relevant atomic density
of bipolarons is well below 1, and the static lattice di-
A.S. Alexandrov, P.E. Kornilovitch / Physics Letters A 299 (2002) 650–655
electric constant is very large in the cuprates, which
justifies the ideal Bose-gas approximation for Tc [6].
Using the experimental λab = 320 nm, λc = 4160 nm,
and RH = 4 × 10−3 cm3 /C (just above Tc ) one obtains Tc = 31 K in striking agreement with the experimental value Tc = 30 K. The recent observation
of the normal state diamagnetism in La2−x Srx CuO4
[26] also confirms the prediction of the bipolaron theory [27]. In this Letter we have not addressed the important issue of the symmetry of the superconducting
order parameter. In the bipolaron theory the symmetry of the pseudogap may be different from that of the
superconducting order parameter. The latter could be
d-wave [23]. Many other features of the bipolaronic
(super)conductor, e.g., the unusual upper critical field,
electronic specific heat, optical and tunneling spectra
match those of the cuprates (for a recent review, see
Ref. [28]).
In conclusion, we have studied a many-polaron
model with strong long-range electron–phonon and
Coulomb interactions. The model shows a rich phase
diagram depending on the ratio of the inter-site Coulomb repulsion and the polaronic (Franck–Condon)
level shift. The ground state is a polaronic Fermi liquid at large Coulomb repulsion, a bipolaronic hightemperature superconductor at intermediate Coulomb
repulsion, and a charge-segregated insulator at weak
repulsion. In the superconducting phase, inter-site
bipolarons are remarkably light leading to a high critical temperature. The model describes many properties
of the underdoped cuprates.
Acknowledgement
This work was supported by EPSRC UK, grant
R46977 (A.S.A.), and by DARPA (P.E.K.).
655
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